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There seems to be quite a bit of jargon around rates, percentages, returns or what not when it comes to the savings accounts. This all seems baffling to me since when you look at a savings account the first thing you are told or see is the APY. Naturally, the simplest of the questions that comes to (at least my) mind is if I put $x$ amount of dollars in a savings account with $m\%$ APY and then get hit by a bus going into a coma for $t$ years having no further interactions with the account, what is the amount of money I'll have in the account after $t$ years?

Here is what I think it is,

$$ f(x) = x\left(1+\frac{m}{100}\right)^t $$

Is this it? Does this quantity have a name that I can look up and find this exact equation?

scribe
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  • This is called the future value of an account. You must consider how interest is compounded. If, for example, interest is compounded monthly, then you get $x\left(1+\frac{m}{1200}\right)^{12t}$. – John Douma Dec 17 '24 at 03:22
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    Is that something that varies bank to bank? What if it compounds continuously? – scribe Dec 17 '24 at 04:25

1 Answers1

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when you look at a savings account the first thing you are told or see is the APY.

The interest rate associated with bank accounts is quite commonly the nominal annual interest rate (though the word "nominal" is almost always dropped) rather than the APY, which is also called the effective annual interest rate.

if I put $x$ amount of dollars in a savings account with $\boldsymbol{m\%}$ APY and have no further interactions with the account, is the amount of money I'll have in the account after $t$ years $f(x) = x\left(1+\frac{m}{100}\right)^t$ ?

Yes.

On the other hand, if $n$ is the number of interest computations per year and $\boldsymbol{r\%}$ is the nominal annual interest rate, then $$f(n,r,x,t) = x\left(1+\frac{r}{100n}\right)^{nt}.$$

If $n>1,$ then $$m>r.$$

ryang
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  • "the interest rate associated with bank accounts is quite commonly the nominal annual interest rate (though the word "nominal" is almost always dropped) rather than the APY, ..."

    Are you saying that what the sites call APY is actually the nominal annual interest rate? Because (as seen in the link), the sites do tell you the APY in case of the savings accounts.

     – scribe Dec 22 '24 at 08:18
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    @scribe No, I didn't say that. However, notice that for the calculation at each interest payment (be it daily or monthly or at whatever frequency), the figure being used is $r$ rather than $m;$ ultimately, what the bank advertises to you depends on where in the world it is operating. Also: many people, even financial-advice websites, conflate the two rates. – ryang Dec 22 '24 at 08:50
  • @scribe If the bank advertises $m$ (the APY), I'd assume that the advertised value has been rounded off; if the bank advertises $r$ (the nominal rate), I'd assume that the advertised value is exact. – ryang Dec 22 '24 at 08:59